H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 18
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sin (28)lJ~n-02-cash [Ln(z+h)I (3,8of)X(A2nr) cosh (bnh)I t is noted that N=5 was used and &.,are defined by-.x~,, x ~ ,and a are neglected with the third harmonicsof order a.$3(3.82)Bz.,(3.god)[h(z+h)lhex& J o ( ~ c w ~cash)(A&)NLa-are introduced.For the cln term to vanish, the conditionsand xl.are satisfied by the assumed form ofThe 2Jaterm contains only the constant termand the second harmonics. There were threeresultant equations which are satisfied by theGalerkin method (or in this case the FourierBessel technique) with the assumed forms ofAAAh,$2, xi.This determinesAn, Aaa, Con,AA4,, D2, in terms of f,. It is noted that theequation resulting from terms independent oftime is satisfied by $o=constant and that theremaining two equations yield six equationsfrom their components in Jo, J2cos (28), J2sin (28).
In the e term, however, only the twofirst harmonic terms were assumed to vanish,treating time derivatives temporarily as constants. The Jl sin 8 and the J1cos 8 components yield a set of four fir~trordernonlineardifferential equations governing f,(i= 1, 2, 3,4) ;namely,cash [ h ( ~ + ~ )(3.80b)]cosh (Xllh)ja(7)(2 [$;) satisfies theplace equation for inviscid irrotational flow aswell as the velocity potential. The coefficientsare to be determined by a combined nonlinearfree surface condition, to the third order.First, by the method of expansion into thesmall parameter c, the coefficient of r"d, Cp, cmust vanish for all times. I n doing so, thetransformationswhereXI=[[&.(z+ h)]-V~Z-KIj2( f !+f:+f:+.fi)+Kz ja(f2 fa-flf4) (3.838)~=~~+~l(j!+f:+fi+fi)+~2f4(fafa-fif4).a=-yj,(j:+j:+fi+fl)-~zfd71(faf1-f j4)(3.83~)95NONLINEAR EFFECTS IN LATERAL. SLOSHINGdf4-vfs(fi+~+fi+f)-~~a(fd7-JS-f~1)motions are now customarily referred to asswirl," and will be discussed in the followingsection of this cha~ter.Now, in order io determine the liquid forceOn the tank (ref.
3.4)~the Pressure is given b yBernoulli's equation in terms of the disturbancepotential, as;I(3.83d)where Kl, Ks, Fl are constants (given in ref.3.19). The values of Kl and Ks depend onmany integrals which have been tabulatedin reference 3.19 with some uncertain errors.'The planar motion is given by the solution -f1=r,fi=fs=fr=oand -y is governed by a cubic equationwhere an arbitrary function of time j(t) can beabsorbed into */at through a redefinition of 4.The *force on the, tank is given by'-Near the k t natural frequency, the instabilityof the planar motion is known to exist.
Themathematical formulation of stability was tointroduce a small disturbance ciek' to the corresponding steady-state amplitude, fi@ andthen determine whether the initial disturbancegrows or decays. If A is positive, the amplitudeof the disturbance grows and the motion isunstable. If X is negative, then the motion isstable. These unstable liquid free surfaceF.=[rrp-ICOSB dBdz]r-awhere tl is the free surface elevation, a is theradius of the rigid tank, and h is the (initial)liquid depth.
F, can be evaluated to the thirdorder, con~ist~entwith Hutton's theory (ref.3.19), except for the contributions due toand xa which were not originally derived.Thus,'In ref. 3.17 thevalues for I:8 are values of I:,/K, is the value of Ko/XL, accordingto com'munications with R. E. Hutton.A?,;I~where-Psin (wt)sinhA11+&[sin (ot) +sin ( 3 4 1-hax msh (Allh) sinh (~~,,h)l+i[sin (ot) +sin ( 3 4 196THE DYNAMIC BEHAVIOR O F LIQUIDSandF~=[- 1''pa cos Bdr]r-a=A2 pa7! rn {: $ sin3((o)[yJl(hlla)]3hlltanh (A11h)---4w2 sin (wt) cos ( 2 4 yJl( h a )-1w sin (wt) l+cos (2wt)+HC ~ n J ~ ( h ? n a ) ] - [ ~7( 2 )hAn-1-Eusin (ut)(3.87a)Qo., 522, are given by Hutton (ref.
3.19). y isthe amplitude of the steady-state part offl, and is governed by a cubic equation den The relationriving &.PA',91 0 )(AI~~)I~](l+cos (2wt))] [ ~ J(Al~a)131A~lIt a d (~II~)J'I)whereAAonJo(A~a>A4, =Concosh ( h h )JIhas been utilized in de-pending on the motion being planar or nonplanar.gA nonlinear theory essentially equivalentto that of Hutton has been derived by Roggeand Weiss (refs. 3.20 and 3.21).Experimental studies of large amplitudeliquid motions in tanks of circular crosssection (cylindrical and spherical) are difficultbecause of the occurrence of swirl modesnear the liquid natural frequencies, as mentionedpreviously and as will be discussed in moredetail in the next section.
Recourse hastherefore been made to the installation ofvertical splitter plates parallel to the directionof excitation to suppress rotational motionsand thus the onset of swirl. Data obtainedfrom such experimental studies (ref. 3.4) willbe summarized in the following section.A typical liquid displacement response curvefor low excitation amplitude is shown in figure3.8. The nonlinear softening characteristicjump phenomenon is clearly demonstrated.Total jorce response data (in the direction ofexcitation) are shown in figure 3.9 for severalvalues of excitation amplitude. All these dataOThrough private communications with R. E.
Hutton,it was learned that I:, in the expressions for 61 andGz should be replaced by Ijl~:,,in order to caloulate7 correctly..r .-'=-:.r*.--. -,--%-?*.w+ - --'--.+-.*%-.>:- ,%%q----..'7-<-** .<--,-.a--.~ . . .*'..,,..*--NONLINEAR EFFECTS IN LATERAL SLOSHING97were to require the development of an improvedtheory, the choice between recommending a"better" third-order theory, or a similarfifth-order theory would not be an easy one.In any event, it may be noted from figure 3.9that the agreement between theory and experiment for the in-phase branches is rather good,for all four values of excitation amplitude.The agreement is not so good for the out-ofphase branches.
I n the vicinity of resonance,the theory departs rather widely from the experimental values and, generally, does not givea good value for the frequency at which thejump occurs.Experimental Data for Nonlinear Liquid Motionsin a Spherical Tank (ref. 3.4)Frequency parameter w'dlgFIGURE3.8.-Liquidfree surface response in halfcylindrical tank dcmomtrating nonlinear charactcristice (ref. 3.4).mere obtained by varying frequency whilemaintaining constant excitation amplitude; thesweep in frequency was made by first strtrtingat a very low value and slowly increasing untilsubstantially above first-mode resonance, andthen sweeping back down to low values.
Thisprocedure revealed jump phenomena veryclearly. The nonlinear softening characteristicof the response curves of figure 3.9 is quite evident.The experimentally determined total forcedata are also compared directly in figure 3.9with calculations from extensions of the analysisof Hutton, as presented earlier in this section.This theory is basically one of third order, buteven then accounts for only certain elementsof the nonlinear aspects of the problem; nevertheless, the theory is quite complex in itsanalytical details, and consequently good agreement with experimental data at and beyondresonance is hardly to be expected.
If oneAs in the case of the circular cylindrical tank,it was found necessary to install a verticalsplitter plate in the spherical tank (parallel tothe direction of excitation) in order to suppressthe liquid swirl mode. Resulting total forceresponse data (amplitude only) for a half-fulltank are shown in figure 3.10. Here again,the response is seen to possess a nonlinearsoftening characteristic, whi.ch is quite sensitiveto excitation amplitude, as had been intimatedsome time ago (ref.
3.7). However, the response curves are not very well defined in theareas of the jumps, as compared with thoseof the cylindrical tank, probably as a consequence of the increased tendency of breakingwaves to form because of the curvature at themean liquid level (breaking wave would notoccur in the cylindrical tank at an equivalentwave amplitude). The breaking waves certainly have some tendency to modify thenormal instability process in the region of thejump frequencies; nevertheless, the jump behtLV<"i:Zrn Seen &ppro-&ate~y &I;eatedfigure 3.10 by the dashed lines.3.4 SWIRL MOTION (ROTARY SLOSHING)General Description of the Liquid MotionMany different experimental studies concerned with lateral sloshing have revealed aninteresting type of liquid instability occurringvery close to the lowest liquid resonant fre-*-98THE DYNM~IC BEHAVIOR OF L I Q ~ D S0020253.03.540Frequency parameter w a d l g455.020253.03.540Frequency parameter w a d l g455.0020253.03.54.0Frequency parameter W' dlg455.020253.03.540Frequency parameter w l d l g455.00hld ' 1.0Excitation-0Increasing frequencyxDecreasing frequencyFIGURE3.9.-Amplitude of liquid force rasponre in half-cyliidrical tank for variour excitation amplitudes (ref.
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